Extrinsic and Intrinsic Visualizations

David W. Henderson

In mathematics classrooms we most often consider shapes and surfaces
extrinsically--looking at the object from the outside. This extrinsic
perspective is inherent in our usual analytic descriptions which depend on an
encompassing 3-space. In addition, when we do visualize or picture an object it
is most often from an external (extrinsic) perspective. However, it is
important for many parts of mathematics to be able to describe and visualize
objects intrinsically--as a being living in (on) the object would experience
it. For example, the great circle on a sphere are, from an extrinsic point of
view, merely circles; however, from the intrinsic point-of-view of a bug whose
universe is the sphere the great circles are experienced as straight. It is in
this setting that spherical geometry most naturally exists. Intrinsic
points-of-view are also important in understanding the possible shapes of our
physical universe and many important parts of differential geometry. I would
also say that intrinsic visualization is important in our understanding of
other people (How is the world experienced and viewed by someone who is
different than I?). I will describe (by as much as possible demonstrating!)
Some of the activities that I have my students do in order to break their
extrinsic habits.